Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{8a^2 + 8a - 448}{7a^3 - 14a^2 - 245a}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ k = \dfrac {8(a^2 + a - 56)} {7a(a^2 - 2a - 35)} $ $ k = \dfrac{8}{7a} \cdot \dfrac{a^2 + a - 56}{a^2 - 2a - 35} $ Next factor the numerator and denominator. $ k = \dfrac{8}{7a} \cdot \dfrac{(a - 7)(a + 8)}{(a - 7)(a + 5)}$ Assuming $a \neq 7$ , we can cancel the $a - 7$ $ k = \dfrac{8}{7a} \cdot \dfrac{a + 8}{a + 5}$ Therefore: $ k = \dfrac{ 8(a + 8)}{ 7a(a + 5)}$, $a \neq 7$